Thick Points for Transient Symmetric Stable Processes
Amir Dembo, Stanford University
Yuval Peres, University of California, Berkeley
Jay Rosen, College of Staten Island, CUNY
Ofer Zeitouni, Technion
Abstract
Let T(x,r) denote the total occupation measure of the
ball of radius r centered at x for a transient symmetric
stable processes of index $b<d$ in $R^d$ and K(b,d)
denote the norm of the convolution with its 0-potential
density, considered as an operator on $L^2(B(0,1),dx)$.
We prove that as r approaches 0, almost surely
$sup_{|x| leq 1} T(x,r)/(r^b|log r|) to b K(b,d)$.
Furthermore, for any $a in (0,b/K(b,d))$, the Hausdorff
dimension of the set of ``thick points'' x for which
$limsup_{r to 0} T(x,r)/(r^b |log r|)=a$,
is almost surely b-a/K(b,d); this is the correct
scaling to obtain a nondegenerate ``multifractal spectrum''
for transient stable occupation measure.
The liminf scaling of T(x,r) is quite different:
we exhibit positive, finite, non-random c(b,d), C(b,d),
such that almost surely
$c(b,d)<sup_x liminf_{r to 0} T(x,r)/r^b<C(b,d)$.
Full text: PDF | PostScript
Copyright for articles published in this journal is retained by the authors, with first publication rights granted to the journal. By virtue of their appearance in this open access journal, articles are free to use, with proper attribution, in educational and other non-commercial settings.
The authors of papers published in EJP/ECP retain the copyright. We ask for the permission to use the material in any form. We also require that the initial publication in EJP or ECP is acknowledged in any future publication of the same article.
Before a paper is published in the Electronic Journal of Probability or Electronic Communications in Probability we must receive a hard-copy of the copyright form. Please mail it to
Philippe Carmona
Laboratoire Jean Leray UMR 6629
Universite de Nantes,
2, Rue de la Houssinière BP 92208
F-44322 Nantes Cédex 03
France
You can also send it by FAX: (33|0) 2 51 12 59 12 to the attention of Philippe Carmona. You can even send a scanned jpeg or pdf of this copyright form to the managing editor ejpecpme@math.univ-nantes.fr. as an attached file.
If a paper has several authors, the corresponding author signs the copyright form on behalf of all the authors.